This is a preprint of: “New symmetric periodic solutions for the Maxwell-Bloch differential system”, Murilo R. Cândido, Jaume Llibre, Clàudia Valls, Math. Phys. Anal. Geom., vol. 22, 16:1–13, 2019. DOI: [10.1007/s11040-019-9313-9] NEW SYMMETRIC PERIODIC SOLUTIONS FOR THE MAXWELL-BLOCH DIFFERENTIAL SYSTEM MURILO R. CÂNDIDO1 , JAUME LLIBRE1 AND CLAUDIA VALLS2 Abstract. We provide sufficient conditions for the existence of a pair of sym- metric periodic solutions in the Maxwell-Bloch differential equations modeling laser systems. These periodic solutions come from a zero-Hopf bifurcation studied using recent results in averaging theory. 1. Introduction and statement of the main result In nonlinear optics the Maxwell–Bloch equations are used to describe laser sys- tems. These equations were obtained by coupling the Maxwell equations with the Bloch equation (a linear Schrödinger like equation which describes the evolution of atoms resonantly coupled to the laser field), see [1]. Now in MathSciNet appear 265 articles related with these equations, see for instance [4, 5, 6, 9, 10, 11]. Recently in [7] it was studied the weak foci and centers of the Maxwell-Bloch system u̇ = − au + v, v̇ = − bv + uw, (1) ẇ = − c(w − δ) − 4uv. For c = 0 the differential system (1) has a singular line {(u, v, w)|u = 0, v = 0}; for c 6= 0 and ac(δ−ab) < 0 the differential system (1) has one equilibrium p0 = (0, 0, δ); and for c 6= 0 and ac(δ − ab) > 0 the differential system (1) has three equilibria p+ = (u∗ , v ∗ , w∗ ), p− = ( − u∗ , −v ∗ , w∗ ) and p0 , where r r ∗ c(δ − ab) ∗ c(δ − ab) ∗ u = ,v =a , w = ab. 4a 4a For a = δ = 0 the differential system has the singular line L = {(u, v, w)|v = 0, w = 0}. The periodic orbits bifurcating from the equilibrium p0 was studied in [2]. Here we complete this study analyzing the periodic orbits which bifurcate form the other two singularities. We define a zero-Hopf equilibrium of a 3-dimensional autonomous differential system as an equilibrium point having two purely conjugate imaginary eigenvalues and a zero eigenvalue. The next result characterizes the zero-Hopf equilibria of system (1) that lies over the singular line L. Proposition 1. Consider a = δ = 0, c = −b, ω ∈ (0, ∞) and   1p 2 q± = ± b + ω 2 , 0, 0 ∈ L. 2 2010 Mathematics Subject Classification. Primary: 34C29 37C27. Key words and phrases. Maxwell-Bloch, averaging theory, periodic solutions, zero-Hopf bifurcations. 1 2 M.R. CÂNDIDO, J. LLIBRE AND C. VALLS The only zero-Hopf equilibria of system (1) in the singular line L are q± . Proposition (1) is proved in Section 2. Perturbing the condition a = δ = 0 the line of singularity L disappears. However the next result shows that there are two equilibrium points in L which produce an isolated periodic solution due to a zero-Hopf bifurcation. Theorem 2. Let ω ∈ (0, ∞), a =ε3 a3 , b =b0 − εc1 − ε2 c2 + ε3 b3 ,   2 3 2a3 b20 c = − b0 + εc1 + ε c2 + ε a3 − b3 + , ω2 a3 ω 2 δ = − ε3 , b0 with (a3 , b0 , b3 , c1 , c2 ) ∈ R5 and ε a small parameter. Then for |ε|6= 0 sufficiently small the Maxwell-Bloch differential system (1) has two symmetric isolated periodic solutions bifurcating from the equilibrium points q± ∈ L when ε = 0 and 2a3 c1 (1 + 6b20 /(ω 2 − 5b20 )) > 0. Theorem 2 is proved in Section 2. 2. The proofs Proof of Proposition 1. Consider q = (ū, 0, 0) ∈ L. In the following we discuss the conditions for q being a zero-Hopf equilibrium point of system (1). The character- istic equation at q is given as  −λ3 + λ2 (−b − c) + λ −bc − 4ū2 = 0. (2) It is easy to check that√equation (2) has the pair of pure imaginary roots ±iω (ω > 0) if and only if ū = ± 21 b2 + ω 2 and c = −b.  Since system (1) is invariant under the transformation (x, y, z) → (−x, −y, z) we proceed the proof only for the point q− . Proof of Theorem 2. Assuming the conditions of Theorem 2 and translating q− to the origin of coordinates, the differential system (1) writes q  a3 u̇ =v + ε3 b20 + ω 2 − 2u , 2 q w v̇ = − b20 + ω 2 − b0 v + uw + εc1 v + ε2 c2 v − ε3 b3 v, q2 ẇ =2v b20 + ω 2 + b0 w − 4uv − εc1 w − ε2 c2 w (3)    2    2 2 2b a3 c1 ω a3 c2 ω + ε3 a3 w − 20 − 1 + ω 2 + b3 w − ε4 − ε5 ω b0 b0 2 2  a3 ω (b3 − a3 ) − 2a3 b0 + ε6 . b0 PERIODIC SOLUTIONS FOR THE MAXWELL-BLOCH SYSTEM 3 In order to write the linear part of system (3) into its Jordan normal form, we do the linear change of variables (u, v, w) → (x, y, z) where ! x 2(b0 y + xω) (u, v, w) = z − , y, − p 2 . ω b0 + ω 2 The differential system (3) becomes  2(ωz − x) 2b20 y + b0 xω + yω 2 c1 ε(2b0 y + xω) c2 ε2 (2b0 y + xω) ẋ = − ωy + p − − 2 ω 2 b0 + ω 2 ω ω  q   ! a3 2b20 + ω 2 (b0 y + xω) 1 2b0 y +ε3 − − a3 ω b20 + ω 2 + b3 +x ω3 2 ω p p p  a3 c1 ωε4 b20 + ω 2 a3 c2 ωε5 b20 + ω 2 a3 ε6 b20 + ω 2 2a3 b20 + ω 2 (a3 − b3 ) + + + , 2b0 2b0 2b0 ω 2(x − ωz)(b0 y + xω) ẏ =ωx + p + c1 yε + c2 yε2 − b3 yε3 , (4) ω b20 + ω 2  2(ωz − x) 2b20 y + b0 xω + yω 2 c1 ε(2b0 y + xω) c2 ε2 (2b0 y + xω) ż = p − − 2 ω 3 b0 + ω 2 ω2 ω2  ε3 b3 ω 2 (2b0 y + xω) − a3 2b30 y + 2b20 xω + b0 yω 2 + ω 4 z + ω4 p p p  4 2 a3 c1 ε b0 + ω 2 5 a3 c2 ε b20 + ω 2 a3 ε6 b20 + ω 2 2a3 b20 + ω 2 (a3 − b3 ) + + + . 2b0 2b0 2b0 ω 2 To study the periodic orbits of system (4) when 0 < |ε| 1, we introduce the cylindrical coordinates x = R cos θ, y = R sin θ and z = Z. Doing this transforma- tion system (4) becomes dR 2b0 R2 cos(θ)(b0 sin(2θ) + ω cos(2θ)) =− p dt ω 2 b20 + ω 2 2b0 RZ(b0 sin(2θ) + ω cos(2θ)) + p + O(ε) ω b20 + ω 2 =R(θ, R, Z),  dθ 2R cos(θ) b20 (− cos(2θ)) + b20 + b0 ω sin(2θ) + ω 2 =ω + p dt ω 2 b20 + ω 2  2Z b20 (− cos(2θ)) + b20 + b0 ω sin(2θ) + ω 2 − p + O(ε) (5) ω b20 + ω 2 =O(θ, R, Z),   dZ 2R2 cos(θ) 2b20 + ω 2 sin(θ) + b0 ω cos(θ) =− p dt ω 3 b20 + ω 2   2RZ 2b20 + ω 2 sin(θ) + b0 ω cos(θ) + p + O(ε) ω 2 b20 + ω 2 =Z(θ, R, Z). Rescaling the variables (R, Z) of system (5) as R = ε2 r, Z = εz, and taking θ as the new independent we obtain the equivalent differential system   dr dz , = εF1 (r, z, θ) + ε2 F2 (r, z, θ) + ε3 F3 (r, z, θ) + ε4 F4 (r, z, θ) + O(ε5 ), (6) dθ dθ 4 M.R. CÂNDIDO, J. LLIBRE AND C. VALLS where Fi (r, z, θ) for i = 1, . . . , 4 are given in Appendix B. Applying Theorem 3 from Appendix A to system (6) we calculate the correspon- dent averaging functions g1 (r, z) = (0, 0), g2 (r, z) = (0, 0),   p  ! π a3 b20 + ω 2 2b20 c1 − 4b0 z b20 + ω 2 + c1 ω 2 − 2b20 r2 g3 (r, z) = 0, p , b0 ω 3 b20 + ω 2 p   4πa3 rz b20 + ω 2 5b20 + ω 2 πr  g4 (r, z) = − a3 c1 b20 + ω 2 ω5 b0 ω 5 (b20 + ω 2 )   2πz 10b40 + 9b20 ω 2 + ω 4 − b20 r2 5b20 + 3ω 2 , b0 ω (b20 + ω 2 ) 5     a3 c1 b20 + ω 2 12b40 + 13b20 ω 2 + 2ω 4 − 2b20 r2 5b20 + 4ω 2 π    + p 2 2b0 c1 r2 5b20 + ω 2 − a3 b20 + ω 2 5 b0 ω b0 + ω 2 ! 3 2 2  2 2 2  4  6b0 c1 + 4z + 2b0 ω −b0 c2 + 2c1 + 6z − c2 ω . Note that the averaged equation g3 (r, z) vanishes over the graph Z = {zα = (α, β(α)) : α ∈ R+ } ⊂ U,   a3 c1 b20 + ω 2 2b20 + ω 2 − 2b20 α2 where β(α) = 3/2 . Furthermore the Jacobian matrix 4a3 b0 (b20 + ω 2 ) of g3 at zα is   0 0 Dg3 (zα ) =  4b0 πα 4a3 π (b20 +ω 2 )  . (7) − 3√ 2 2 − ω3 ω b0 +ω  From (7) we have that ∆α = −4a3 π b20 + ω 2 /ω 3 6= 0. This verifies the conditions (i) and (ii) of Theorem 3. Thus we calculate the function f (x) and we get   πb0 α ω 2 α2 − 2a3 b20 c1 − 2a3 c1 ω 4 − 5b20 α2 f (α) = . ω 5 (b20 + ω 2 ) It is easy to check that f (α) has the positive zero s     ∗ 6b20 6b20 α = ω 2a3 c1 + 1 if 2a c 3 1 + 1 > 0. ω 2 − 5b20 ω 2 − 5b20 Moreover, α∗ is a simple zero because f 0 (α∗ ) = 4πa3 b0 c1 /ω 3 6= 0. Thus the result follows from applying Theorem 3. This conclude the proof.  Appendix A: Averaging theory We consider differential systems of the form e x, ε), (8) ẋ = F0 (t, x) + εF1 (t, x) + ε2 F2 (t, x) + ε3 F3 (t, x) + ε4 F4 (t, x) + ε5 F(t, with x in some open subset Ω of Rn , t ∈ [0, ∞), ε ∈ [−ε0 , ε0 ]. We assume Fi and e for all i = 1, 2, 3, 4 are T –periodic in the variable t. Let x(t, z, 0) be the solution F of the unperturbed system ẋ = F0 (t, x), PERIODIC SOLUTIONS FOR THE MAXWELL-BLOCH SYSTEM 5 such that x(0, z, 0) = z. We define M (t, z) the fundamental matrix of the linear differential system ∂F0 (t, x(t, z, 0)) ẏ = y, ∂x such that M (0, z) is the n × n identity matrix. The displacement map of system (8) is defined as d(z, ε) = x(T, z, ε) − z. (9) In order to have d(z, ε) well defined we assume that for |ε|6= 0 sufficiently small the following hypothesis holds: (H) there exists an open set U ⊂ Ω such that for all z ∈ U the solution x(t, z, ε) is defined on the interval [0, t(z,ε) ) with t(z,ε) > T . This hypothesis is always satisfied when the unperturbed system has a manifold of T -periodic solutions. The standard method of averaging for finding periodic solutions consists in write the displacement map (9) in power series of ε as follows d(z, ε) = g0 (z) + εg1 (z) + ε2 g2 (z) + ε3 g3 (z) + ε4 g e(z, ε), Where for i = 0, 1, 2, 3, 4 we have yi (T, z) gi (z) = M (T, z)−1 , i! being y0 (t, z) =x(t, z, 0) − z, Z t y1 (t, z) =M (t, z) M (τ, z)−1 F1 (τ, x(τ, z, 0))dτ, 0 Z " t ∂F1 y2 (t, z) =M (t, z) M (τ, z)−1 2F2 (τ, x(τ, z, 0)) + 2 (τ, x(τ, x, 0))y1 (τ, z) 0 ∂x # ∂ 2 F0 2 + (τ, x(τ, z, 0))y1 (τ, z) dτ, ∂x2 Z " t ∂F2 y3 (t, z) =M (t, z) M (τ, z)−1 6F3 (τ, x(τ, z, 0)) + 6 (τ, x(τ, z, 0))y1 (τ, z) 0 ∂x 2 ∂ F1 ∂F1 +3 (τ, x(τ, z, 0))y1 (τ, z)2 + 3 (τ, x(τ, z, 0))y2 (τ, z) ∂x2 ∂x # ∂ 2 F0 ∂ 3 F0 3 +3 (τ, x(τ, z, 0))y1 (τ, z) y2 (τ, z) + (τ, x(τ, z, 0))y1 (τ, z) dτ, ∂x2 ∂x3 Z t " −1 ∂F3 y4 (t, z) =M (t, z) M (τ, z) 24F4 (τ, x(τ, z, 0)) + 24 (τ, x(τ, x, 0))y1 (τ, z) 0 ∂x ∂ 2 F2 ∂F2 + 12 (τ, x(τ, z, 0))y1 (τ, z)2 + 12 (τ, x(τ, z, 0))y2 (τ, z) ∂x2 ∂x ∂ 2 F1 ∂ 3 F1 + 12 (τ, x(τ, z, 0))y 1 (τ, z) y 2 (τ, z) + 4 (τ, x(τ, z, 0))y1 (τ, z)3 ∂x2 ∂x3 ∂F1 ∂ 2 F0 +4 (τ, x(τ, z, 0))y3 (τ, z) + 3 (τ, x(τ, z, 0))y2 (τ, z)2 ∂x ∂x2 ∂ 2 F0 +4 (τ, x(τ, z, 0))y1 (τ, z) y3 (τ, z) ∂x2 6 M.R. CÂNDIDO, J. LLIBRE AND C. VALLS # ∂ 3 F0 ∂ 4 F0 +6 (τ, x(τ, z, 0))y1 (τ, z)2 y2 (τ, z) + 4 (τ, x(τ, z, 0))y1 (τ, z) dτ. ∂x3 ∂x4 The functions g1 , g2 , g3 and g4 will be called here the averaged functions of order 1, 2, 3 and 4 respectively of system (8). We say that system (8) has a periodic solution bifurcating from the point z0 if there exists a branch of solutions x(t, z(ε), ε) such that the displacement function satisfies d(z(ε), ε) = 0 and z(0) = z0 . Let π : Rm × Rn−m → Rm and π ⊥ : Rm × Rn−m → Rn−m denote the projections onto the first m coordinates and onto the last n − m coordinates, respectively. For a point z ∈ U we also consider z = (a, b) ∈ Rm × Rn−m . Consider the graph Z = {zα = (α, β(α)) : α ∈ V } ⊂ U such that m < n, V is an open set of Rm and β : V → Rn−m is a C 4 function. The next theorem provides sufficient conditions for the existence of periodic solutions of the differential system (8) when the set Z is a continuum of zeros to the first non vanishing averaged equation. Theorem 3. Let r ∈ {0, 1, 2, 3} such that r is the first subindex such that gr 6≡ 0. In addition to hypothesis (H) assume that (i) the averaged function gr vanishes on Z. That is gr (zα ) = 0 for all α ∈ V , and (ii) the Jacobian matrix   Λα Γα Dgr (zα ) = , B α ∆α where Λα = Da πgr (zα ), Γα = Db πgr (zα ), Bα = Da π ⊥ gr (zα ) and ∆α = Db π ⊥ gr (zα ), satisfies that det(∆α ) 6= 0 for all α ∈ V . We define the function f (α) = −Γα ∆−1 ⊥ α π gr+1 (zα ) + πgr+1 (zα ), Then the following statements hold. (a) If there exists α∗ ∈ V such that f (α∗ ) = 0 and det (Df (α∗ )) 6= 0, for |ε|6= 0 sufficiently small, then there is an initial condition z(ε) ∈ U such that z(0) = zα∗ and the solution x(t, z(ε), ε) of system (8) is T -periodic. For a proof of Theorem 3 see [4]. The ideas of the proof were first presented in [8]. Appendix B: The functions Fi (r, z, θ) for i = 1, 2, 3, 4. In the following functions we take S = sin θ, C = cos θ, S2 = sin(2θ), C2 = cos(2θ) and C3 = cos(3θ).  q  ! 2r 2 2 2  F1 (r, z, θ) = p 2b0 z − c1 b20 + ω 2 (b0 S2 + ωC2 ) − a3 ω b0 + ω C, 0 2ω 2 b20 + ω 2   q  1 1 2 2  2 + ω 2 − 2b z F2 (r, z, θ) = ωS a3 ω b0 + ω + 4rC c1 b 0 0 4ω 4 r (b20 + ω 2 ) PERIODIC SOLUTIONS FOR THE MAXWELL-BLOCH SYSTEM 7 q  q    + 2b0 c1 r b20 + ω 2 +2b0 rC2 2b0 z − c1 b20 + ω 2 − 4rz b20 + ω 2   q    a3 ω 2 b20 + ω 2 C − 2r 2b0 z − c1 b20 + ω 2 (b0 S2 + ωC2 ) 2ω   + p C b20 a3 c1 ω 3 − 2r2 ω + a3 c1 ω 5 − 4b30 r2 S2 b0+ b20 ω2  q  ! 2 r −2b0 rω c2 b0 + ω 2 (b0 S2 + ωC2 ) + b0 rC3 , p b20 + ω 2 ω3   q   q  ! 2 2 2 2 2 2 2S −b0 c1 b0 + ω + 2b0 z + ω z + ωC 2b0 z − c1 b0 + ω ,  q  1 1 2 + ω 2 (b S F3 (r, z, θ) = p 2r 2b 0 z − c 1 b0 0 2 4ω 5 2r2 ω b20 + ω 2 !   1   2 2 2 + ωC2 ) − a3 ω b0 + ω C 2 2 ωS a3 ω b20 + ω 2 b0 + ω  q ! q + 4rC c1 b20 + ω 2 − 2b0 z + 2b0 c1 r b20 + ω 2 + 2b0 rC2  q  2  2rω 2b0 z − c1 b20 + ω2 − 4rz b20 +ω 2 − p b0 b20 + ω 2   q  −ωS a3 c1 ω 2 b20 + ω 2 − 4b0 c2 r b20 + ω 2   (b0 S + ωC)) + 4b0 r2 2b20 + ω 2 S 2 C + 8b20 r2 ωSC 2 + 4b0 r2 ω 2 C 3 1  + 2 2 ωC 4b20 r2 C2 − a3 c1 ω 2 b20 + ω 2 + 8b30 r2 SC 2 b0 r (b0 + ω ) q   +2b0 c2 rω b20 + ω 2 (b0 S2 + ωC2 ) ωS a3 ω b20 + ω 2  q  q +4rC c1 b20 + ω 2 − 2b0 z + 2b0 c1 r b20 + ω 2 + 2b0 rC2  q    2ω 2b0 z − c1 b20 + ω 2 − 4rz b20 + ω 2 + b0   q  −4a3 b30 rωC 2 + a3 C c2 ω 4 b20 + ω 2 − 4b40 rS ! 2  −b0 rω ((a3 − 2b3 )(b0 S2 + ωC2 ) + a3 ω) ,   q  1 ω p a3 ω 3 b20 c1 − 2b0 z b20 + ω 2 + c1 ω 2 2ω 5 b20 + ω 2 b0 q  q − 2b0 c2 rω 2 b20 + ω 2 C − 4b0 r b0 c2 ω b20 + ω 2 S   1 + r 2b20 + ω 2 SC + b0 rωC 2 − 2S + ω2 b20  q   q ! −b0 c1 b20 + ω 2 + 2b20 z + ω 2 z + ωC 2b0 z − c1 b20 + ω 2 8 M.R. CÂNDIDO, J. LLIBRE AND C. VALLS    q   ωS a3 ω b20 + ω 2 + 4rC c1 b20 + ω 2 − 2b0 z q  q  !! 2 2  + 2b0 c1 r b20 2 + ω 2 + 2b0 rC2 2b0 z − c1 b0 + ω 2 − 4rz b0 + ω ,    q 1  1 F4 (r, z, θ) = p − 2c3 δ3 b20 +ω 2 Cω + 2 2 2c3 rωC + c2 δ3 b20 + ω 2 4ω 4 b20 + ω 2 r   −2b0 (b3 − c3 )rS) C + 2b3 rωS 2 −4rzω 2 C 2 − S b0 δ3 b20 + ω 2  q   q  2 2 2 2 2 2 +2rωC 4b0 z + (b1 − c1 ) b0 + ω + 2rS 4zb0 + (b1 − c1 ) b0 + ω b0 + 2zω   q  1 + 2 3 −4b0 r ωC − 2r 4rSb0 + c2 ω b0 + ω C 2 − ω −4b0 r2 S 2 2 3 2 2 2 2 2r ω q  q  2 2 2  2 2 2 2 −2b0 (b2 − c2 )r b0 + ω S + c1 δ3 b0 + ω C − 2b2 rω b0 + ω S 2    q  1 2 2  4rzω C + S b0 δ3 b20 +ω 2 2 + 2rωC 4b0 z + (b1 − c1 ) b0 + ω 2 b20 + ω 2  q  !2 2rω +2rS 4zb20 + (b1 − c1 ) b20 + ω 2 b0 + 2zω 2 +p 2 −4r2 ω 2 C 3 b0 + ω 2  q  − 8b0 r2 ωSC 2 − S 2 2b20 + ω 2 S2 r2 − 2b0 (b2 − c2 )ω b20 + ω 2 Sr q    q   1 +c1 δ3 ω b20 + ω 2 + (b2 − c2 )rω 2 S2 b20 + ω 2 − 3 4 2rω 2b0 z − c1 b20 + ω 2 4r ω   q   C 2 + b0 δ3 b20 + ω 2 + 2rS 4b0 z + (b1 − c1 ) b20 + ω 2 C − 2rω (2b0 z q    q  +b1 b20 + ω 2 S 2 −4r2 S 2(b3 − c3 )rωC + 2b0 (b3 − c3 )rS − c2 δ3 b20 + ω 2 ω 4    q 2rω 2 2 2 2  2 + 2 4rzω C + S b δ 0 3 b0 + ω + 2rS 4zb0 + (b 1 − c1 ) b20 + ω 2 b0 b0 + ω 2  q   +2zω 2 + rωS2 4b0 z + (b1 − c1 ) b20 + ω 2 4r2 ω 2 C 3 + 8b0 r2 ωSC 2 + ωS  q   q   c1 δ3 b20 + ω 2 − 2b0 (b2 − c2 )r b20 + ω 2 S + r (c2 − b2 ) b20 + ω 2 ω 2 + 2r 2b20 + ω 2   1  S S2 + p 2 −4rzω 2 C 2 − S b0 δ3 b20 + ω 2 + 2rωC (4b0 z b0 + ω 2 q   q   1 +(b1 − c1 ) b20 + ω 2 + 2rS 4zb20 + (b1 − c1 ) b20 + ω 2 b0 + 2zω 2 b20 + ω 2    q   4rzω 2 C 2 + S b0 δ3 b20 + ω 2 + 2rωC 4b0 z + (b1 − c1 ) b20 + ω 2  q  !2 2rω +2rS 4zb20 + (b1 − c1 ) b20 + ω 2 b0 + 2zω 2 +p 2 −4r2 ω 2 C 3 b0 + ω 2  q  − 8b0 r2 ωSC 2 − S 2 2b20 + ω 2 S2 r2 − 2b0 (b2 − c2 )ω b20 + ω 2 Sr + PERIODIC SOLUTIONS FOR THE MAXWELL-BLOCH SYSTEM 9 q    c1 δ3 ω b20 + ω 2 + (b2 − c2 )rω 2 S2 b20 + ω 2 , 1  1 − − 2a3 rωC + 2c3 rωC − 2b0 b3 rS + 2b0 c3 rS + 2ω 3 2rω (b20 + ω 2 ) 3   q  2 2 2 2 2 2  2 4b0 r ωC + 2r (c2 − a2 ) b0 + ω ω + 2r 2b0 + ω S C  q  q  2 2 2 2 2 2 +ω c1 δ3 b0 − 2(b2 − c2 )r b0 + ω Sb0 + ω c1 δ3 + 2a3 z b0 + ω    q  2 2 2 2  2 4rzω C + S b0 δ3 b0 + ω + 2rωC 4b0 z + (b1 − c1 ) b0 + ω 2  q  2 2 1 2 +2rS 4zb0 + (b1 − c1 ) b0 + ω b0 + 2zω 2 − p a3 ω 4 4r ω b20 + ω 2 2 4 q  q  + a3 b20 ω 2 + b0 δ3 ω 2 − 2a2 z b20 + ω 2 ω 2 + 2rC 2b0 z − c1 b20 + ω 2 ω + b30 δ3  q   1 +2rS 4zb20 + (b1 − c1 ) b20 + ω 2 b0 + 2zω 2 4rzω 2 C 2 + S b20 + ω 2   q   b0 δ3 b20 + ω 2 + 2rωC 4b0 z + (b1 − c1 ) b20 + ω 2 + 2rS 4zb20 q 2 2 2rω 2 2 +(b1 − c1 ) b0 + ω b0 + 2zω +p 2 −4r2 ω 2 C 3 − 8b0 r2 ωS b0 + ω 2  q  C 2 − S 2 2b20 + ω 2 S2 r2 − 2b0 (b2 − c2 )ω b20 + ω 2 Sr q  p 2 2  2 2 a2 b20 + ω 2 +c1 δ3 ω b0 + ω + (b2 − c2 )rω S2 b0 + ω 2 + 8r3 ω 4   q  −4r S 2(b3 − c3 )rωC + 2b0 (b3 − c3 )rS − c2 δ3 b0 + ω ω 4 2 2 2 2rω  + 4rzω 2 C 2 + S b0 δ3 b20 + ω 2 + 2rS 4zb20 b20 +ω 2 q   q  +(b1 − c1 ) b20 + ω 2 b0 + 2zω 2 + rωS2 4b0 z + (b1 − c1 ) b20 + ω 2  4r2 ω 2 C 3 + 8b0 r2 ωSC 2 + ωS c1 δ3 b20 + ω 2 − 2b0 (b2 − c2 ) q   q    r b20 + ω 2 S + r (c2 − b2 ) b20 + ω 2 ω 2 + 2r 2b20 + ω 2 S S2 1     +p −4rzω 2 C 2 − S b0 δ3 b20 + ω 2 + 2rωC 4b0 z b20 + ω 2 q   q  +(b1 − c1 ) b20 + ω 2 + 2rS 4zb20 + (b1 − c1 ) b20 + ω 2 b0 + 2zω 2     1  2 2 4rzω 2 C 2 + S b0 δ3 b20 + ω 2 + 2rωC 4b0 z b0 + ω q   q 2 2 2 2 2 2 2 +(b1 − c1 ) b0 + ω + 2rS 4zb0 + (b1 − c1 ) b0 + ω b0 + 2zω 2rω  +p −4r2 ω 2 C 3 − 8b0 r2 ωSC 2 − S 2 2b20 + ω 2 S2 r2 b20 + ω2 10 M.R. CÂNDIDO, J. LLIBRE AND C. VALLS q  q   −2b0 (b2 − c2 )ω b20 + ω 2 Sr + c1 δ3 ω b20 + ω 2 + (b2 − c2 )rω 2 S2 b20 + ω 2 q !  + c2 δ 3 b20 + ω 2 . Acknowledgements The first author is supported by CNPq 248501/2013-5 and FAPESP 2018/07344- 0. The second author is partially supported by the Ministerio de Economı́a, Indus- tria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013. References [1] F. T. Arecchi, Chaos and generalized multistability in quantum optics, Physica Scripta 9 (1985), 85–92. [2] M. R. Cândido, J. Llibre and D. D. Novaes, Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction, Nonlinearity 30 (2017) 35–60. [3] M. R. Cândido and J. Llibre, Stability of periodic orbits in the averaging theory: Applica- tions to Lorenz and Thomas’ differential systems, Int. J. Bifurcation and Chaos 28 (2018), 1830007–pp 14. [4] I. Casu and C. Lazureanu, Stability and integrability aspects for the Maxwell-Bloch equa- tions with the rotating wave approximation, Regul. Chaotic Dyn. 22 (2017), no. 2, 109–121. [5] C. Lazureanu, The real-valued Maxwell-Bloch equations with controls: from a Hamilton- Poisson system to a chaotic one, Int. J. Bifur. Chaos Appl. Sci. Engrg. 27 (2017), no. 9, 1750143, 17 pp. [6] C. Lazureanu, On a Hamilton-Poisson approach of the Maxwell-Bloch equations with a control, Math. Phys. Anal. Geom. 20 (2017), no. 3, Art. 20, 22 pp. [7] L. Liu, O. O. Aybar, V. G. Romanovski and W. Zhang, Identifying weak foci and centers in Maxwell–Bloch system, J. Math. Anal. Appl. 430 (2015), 549–571. [8] J. Llibre and D. D. Novaes, Improving the averaging theory for computing periodic solu- tions of the differential equations, Zeitschrift für angewandte Mathematik und Physik (2015) 1401–1412. [9] D.W. Zuo, Modulation instability and breathers synchronization of the nonlinear Schrödinger Maxwell–Bloch equation, Appl. Math. Lett. 79 (2018), 182–186. [10] L. Wang, Z.Q. Wang, W.R. Sun, Y.Y. Shi, M. Li and M. Xu, Dynamics of Peregrine combs and Peregrine walls in an inhomogeneous Hirota and Maxwell-Bloch system, Commun. Nonlinear Sci. Numer. Simul. 47 (2017), 190–199. [11] J. Wei, X. Wang and X. Geng, Periodic and rational solutions of the reduced Maxwell-Bloch equations, Commun. Nonlinear Sci. Numer. Simul. 59 (2018), 1–14. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: candidomr@mat.uab.cat Email address: jllibre@mat.uab.cat 2 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal Email address: cvalls@math.ist.utl.pt